Compensated molds for manufacturing ion exchange strengthened, 3D glass covers

ABSTRACT

Methods for compensating for the warp exhibited by three-dimensional glass covers as a result of ion exchange strengthening are provided. The methods use a computer-implemented model to predict/estimate changes to a target three-dimensional shape for the 3D glass cover as a result of ion exchange strengthening. The model includes the effects of ion exchange through the edge of the 3D glass cover. In an embodiment, the inverse of the predicted/estimated changes is used to produce a compensated (corrected) mold which produces as-molded parts which when subjected to ion exchange strengthening have shapes closer to the target shape than they would have had if the mold had not been compensated (corrected).

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority under 35 U.S.C. §119 ofU.S. Provisional Application Ser. No. 61/820,318 filed on May 7, 2013the content of which is relied upon and incorporated herein by referencein its entirety.

FIELD

This disclosure relates to three-dimensional glass covers (3D glasscovers) for electronic devices, such as, mobile or handheld electronicdevices. More particularly, the disclosure relates to three-dimensionalglass covers that have been ion exchange strengthened (IOX strengthened)and to the molds used to make such glass covers.

BACKGROUND

FIG. 1 shows a representative, non-limiting, shape for a 3D glass cover(also known in the art as a “3D cover glass”), which can be used with anelectronic device, such as, a telephone, television, tablet, monitor, orthe like. As shown in this figure, 3D glass cover 100 includes: (i) aplanar central portion 101, (ii) a perimeter portion 102, and (iii) aperimeter edge 103.

Planar central portion 101 is flat or nearly flat, i.e., its radius ofcurvature is at least 300 millimeters. Perimeter portion 102 extends outof the plane of the planar central portion 101, thus providing the glasscover with an overall three-dimensional shape, as opposed to atwo-dimensional shape. Although as shown in FIG. 1, perimeter portion102 completely surrounds central portion 101, in some embodiments, theperimeter portion can extend around only a portion of the centralportion, e.g., for a glass cover having a rectangular shape, less thanall four sides of the glass cover can include a perimeter portion, e.g.,two sides can have a perimeter portion and the other two sides can beflat or substantially flat. Likewise, to be three-dimensional, a glasscover in the form of a disc or saucer only needs to have a portion ofits flat or nearly flat central portion transition into a perimeterportion which extends out of the plane of the flat or nearly flatcentral portion.

As will be evident, the shape of a 3D glass cover can vary widelydepending on the desires of the designer of the device with which the 3Dglass cover will be used. Thus, the 3D glass cover can have a variety ofoverall shapes and can include central portions and perimeter portionsof various sizes and shapes, and can employ transitions of variousconfigurations between the central and perimeter portions.Commonly-assigned U.S. application Ser. No. 13/774,238 entitled “CoverGlass Article” filed Feb. 22, 2013, published as U.S. Patent ApplicationPublication No. 2013/0323444, the contents of which are incorporatedherein by reference, provides various representative dimensions for 3Dglass covers, as well as descriptions of typical applications for thecovers. The molding technology disclosed herein can be used with 3Dglass covers of these types, as well as other types now known orsubsequently developed.

The transverse dimension (thickness) of perimeter edge 103 correspondsto the thickness of the glass from which the glass cover is made, whichis typically less than 1 millimeter, e.g., 0.8 millimeters or less.Because of this small thickness, prior to the present disclosure, it hadbeen believed that stress changes at the edge could be ignored inpredicting changes in the overall shape of a 3D glass cover as a resultof ion exchange (IOX) strengthening. In particular, on an area basis,the perimeter edge of a typical glass cover amounts to less than abouttwo percent of the overall area of the part. Hence, the number of ionsexchanged through the edge is only a small fraction of the total numberof ions exchanged, thus making it reasonable to assume that relative tothe total number of ions exchanged, those few ions would have littleeffect on the structural behavior of the part.

In fact, in accordance with the present disclosure, it has beensurprisingly found that ion exchange at the edge not only has asubstantial effect on the overall shape, but its effect is in many casesgreater than the effect of ion exchange on the rest of the part. Thus,although only a small number of ions move through the edge relative tothe total number of ions that move through the surfaces of the rest ofthe part, those edge-traversing ions are critical to the shape changesexhibited by 3D glass covers as a result of IOX strengthening. In termsof commercial value, this discovery permits manufacturers of 3D glasscovers to effectively and efficiently meet tolerance requirements ofcustomers for those covers. In particular, as detailed below, it allowsmanufacturers of 3D glass covers to produce molds for making thosecovers which accurately compensate for the changes in shape which thecover will exhibit when IOX strengthened. The technology thus,represents a valuable contribution to the ability of designers to createaesthetically pleasing designs for 3D glass covers and the manufacturersof the covers to accurately produce the shapes envisioned by thedesigners.

SUMMARY

In accordance with a first aspect, a method is disclosed of making aglass cover (100), said glass cover (100) having a targetthree-dimensional shape which comprises a planar central portion (101)and a perimeter portion (102) which (i) borders at least part of theplanar central portion (101) and (ii) extends out of the plane of theplanar central portion (101) to provide the glass cover (100) with threedimensionality, said perimeter portion (102) having a perimeter edge(103), said method comprising:

(I) providing a mold (200) for forming the glass cover (100), said mold(200) having a molding surface (208);

(II) producing the glass cover (100) using the mold (200) of step (I);and

(III) ion exchange strengthening the glass cover (100) produced in step(II);

wherein the shape of the molding surface (208) of the mold (200) of step(I) is based at least in part on a computer-implemented model whichpredicts/estimates changes to the target three-dimensional shape as aresult of the ion exchange strengthening of step (III), saidcomputer-implemented model including the effects of ion exchange throughthe perimeter edge (103).

Optionally, the glass cover can be annealed to relieve residual thermalstress between steps (II) and (III).

In accordance with a second aspect, a computer-implemented method isdisclosed for predicting/estimating changes in the shape of athree-dimensional glass cover (100) after ion exchange strengthening,said glass cover (100) comprising a planar central portion (101) and aperimeter portion (102) which (i) borders at least part of the planarcentral portion (101) and (ii) extends out of the plane of the planarcentral portion (101) to provide the glass cover (100) with threedimensionality, said perimeter portion (102) having a perimeter edge(103), said method comprising employing a boundary condition at theperimeter edge (103) which permits ion permeation through the edge so asto model the effects of ion exchange through the perimeter edge (103) onthe shape of the glass cover (100).

In embodiments of the above methods, ion diffusion is treated as thermaldiffusion and the boundary condition at the perimeter edge permits heatflow through the edge.

The reference numbers used in the above summaries of the aspects of theinvention are only for the convenience of the reader and are notintended to and should not be interpreted as limiting the scope of theinvention. More generally, it is to be understood that both theforegoing general description and the following detailed description aremerely exemplary of the invention and are intended to provide anoverview or framework for understanding the nature and character of theinvention.

Additional features and advantages of the invention are set forth in thedetailed description which follows, and in part will be readily apparentto those skilled in the art from that description or recognized bypracticing the invention as exemplified by the description herein. Theaccompanying drawings are included to provide a further understanding ofthe invention, and are incorporated in and constitute a part of thisspecification. It is to be understood that the various features of theinvention disclosed in this specification and in the drawings can beused in any and all combinations.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view of a representative 3D glass cover.

FIG. 2 is a schematic, cross-sectional diagram illustrating arepresentative mold for producing a 3D glass cover.

FIG. 3 shows a cross-sectional slice of an example saucer-shaped part.The “edge” of the part is indicated by arrows.

FIG. 4 shows a cross-sectional slice of a bent shape for the part ofFIG. 3. The x-axis and y-axis numbers are arbitrary in this figure.

FIG. 5 is a schematic diagram showing an original (solid lines) and awarped (dashed lines) axisymmetric shape.

FIG. 6 is a schematic diagram illustrating a geometry for which ananalytic solution for ion diffusion can be obtained.

FIG. 7 is a schematic diagram showing finite element analysis resultsfor the geometry shown in FIGS. 3 and 4. The straight lines show wherethe glass was before ion exchange.

FIG. 8 is a schematic diagram showing finite element analysis resultsfor the geometry shown in FIGS. 3 and 4. The straight lines show wherethe glass was before ion exchange.

FIG. 9 is a schematic diagram showing finite element analysis resultsfor the geometry shown in FIGS. 3 and 4. In particular, this figuremagnifies the cover's edge to show that only this part of the cover wasion exchanged. The straight lines show where the glass was before ionexchange.

FIG. 10 is a schematic, cross-sectional view of a saucer-shaped partidentifying various geometric variables.

FIG. 11 is a schematic diagram showing finite element analysis resultsfor six cases, where α=10° for the left-hand column and α=90° for theright-hand column. The straight lines show where the glass was beforeion exchange.

FIG. 12 is a graph showing warp of the saucer part of FIG. 10 vs. angleα. At α=90° the height h of FIG. 10 is 7.7 mm.

FIG. 13 is a graph showing warp of the saucer part of FIG. 10 vs. angleβ.

FIG. 14 is graph showing warp of the saucer part of FIG. 10 vs. thelength x of the out-of-plane, perimeter portion of the saucer.

FIG. 15 is graph showing warp of the saucer part of FIG. 10 vs. thelength of the flat central portion of the saucer.

FIG. 16 is a schematic, perspective view of a portion of arepresentative 3D glass cover illustrating representative meshing overthe major surfaces of the glass cover suitable for use in predicting thewarp of the part as a result of IOX strengthening.

FIG. 17 is a schematic, perspective view of a portion of arepresentative 3D glass cover illustrating representative meshing overthe edge of a 3D glass cover whose IOX-warp behavior is to bepredicted/estimated.

FIG. 18 is a flowchart illustrating a mold contour correction method inaccordance with an embodiment of the disclosure.

FIG. 19 is a schematic diagram illustrating the importance of includingthe edge of a 3D glass cover when modeling warp of the cover during IOXstrengthening.

FIG. 20 is a graph illustrating IOX warp with respect to CAD shape(target shape) in the curved region (perimeter portion) of a 3D glasscover.

FIG. 21 is a graph illustrating IOX warp with respect to CAD shape(target shape) in the flat region (central portion) of a 3D glass cover.

FIG. 22 is a graph of measured data for a 3D glass cover as-molded(curve 108) and post-IOX (curve 109) where the mold was uncorrected.

FIG. 23 is a graph of measured data for a 3D glass cover as-molded(curve 110) and post-IOX (curve 111) using the same mold as FIG. 22 butafter correction in accordance with the present disclosure.

The warp shown in FIGS. 4-5, 7-9, 11, and 19 is not drawn to scale buthas a magnified y-axis scale for purposes of illustration.

DETAILED DESCRIPTION

As discussed above, the present disclosure is concerned with methods ofproducing IOX-strengthened, 3D glass covers having a shape thatcorresponds closely to that specified by the designer of the cover (thetarget shape; also known as the “CAD shape” in cases where the shape isspecified through a CAD drawing). The disclosure also relates to methodsfor designing molds having mold surfaces which compensate for (correctfor) the changes in shape which a 3D glass cover undergoes when IOXstrengthened.

In particular, after ion exchange, a 3D glass cover warps in the rangeof 10-150 microns depending upon the part shape. This happens because ofdilatation in IOX: smaller ions, e.g., sodium ions, are replaced bylarger ions, e.g., potassium ions, in the outer ˜40 to ˜100 microns ofglass thickness. This causes the glass's dimensions to increase. In flatglass, this dimensional growth in IOX is about 0.04%. As will bediscussed below, in glass having a 3D shape, the behavior of the glassis much more complicated than a simple dimensional growth.

Shape deviation as a result of IOX warp is not desirable as customerspecifications are typically ±100 microns. In order to compensate forthis IOX warp, in accordance with the present disclosure, mold contourcorrections are used which cause the as-molded part to deviate from thetarget shape in a manner such that after IOX strengthening, the shape ofthe part is closer to the target shape than it would have been withoutthe corrections. That is, the as-molded part is moved away from thetarget shape so that the IOX strengthened part will be closer to thatshape.

Because IOX warp depends on the details of the overall shape of theglass cover, as well as on details of the shape and thickness of thecover's edge, in general, individualized mold correction values areneeded for each 3D glass cover that is to be produced. In accordancewith an aspect of the disclosure, these correction values are obtainedby transforming the IOX warp problem into a thermal diffusion problem,thus allowing the IOX problem to be solved using commercially-availablesoftware, e.g., ANSYS® software sold by ANSYS Inc., 275 TechnologyDrive, Canonsburg, Pa. 15317, USA, which employs thoroughly-tested,state-of-art finite element and graphical display techniques. Also,target shapes and, in particular, CAD-formatted target shapes, can bereadily inputted to such commercially-available software. In practice,using the technology disclosed herein, mold contour corrections can berapidly developed without the need for repeated iterative changes tophysical molds. Indeed, in many cases, a single iteration will besufficient to reduce deviation from CAD after IOX to approximately ±10microns, thus allowing the 3D glass covers to meet customerspecifications.

In some embodiments, the 3D glass cover is made from a 2D glass sheetusing a thermal reforming process such as described in U.S. PatentApplication Publications Nos. 2010/0000259 and 2012/0297828, bothincorporated herein by reference. In some embodiments, the 2D glasssheet is made by a fusion process, although 2D glass sheets made byother processes, such as by float or rolling processes, may also beused.

FIG. 2 is a schematic, cross-sectional diagram illustrating arepresentative mold suitable for use in a thermal reforming process ofthe type disclosed in the above-referenced patent applications. In thisfigure, mold 200 includes a mold body 202 having a top surface 206 and acavity 204. The cavity is open at the top surface 206 and its bottomcomprises molding (shaping) surface 208. Molding surface 208 has asurface profile which, in accordance with the present disclosure, iscorrected to compensate for IOX warping. As can be appreciated, theprofile of molding surface 208 will vary from that shown in FIG. 2depending on the specifics of the 3D glass cover that is to be made.

As shown in FIG. 2, mold body 202 can include one or more slots and/orholes 210 (hereinafter referred to as “apertures”) extending from thebottom surface 215 of the mold body to the molding surface. Apertures210 are arranged to provide communication between the exterior of themold and the molding surface. In one example, the apertures are vacuumapertures. That is, the apertures can be connected to a vacuum pump orother device (not shown) for providing vacuum to the cavity 204 throughthe molding surface 208.

FIG. 2 also shows a flat glass plate 218 having a portion 220 locatedover cavity 204. Briefly, in forming a 3D glass cover using a mold ofthe type shown in FIG. 2, heat is applied to plate 218 so that it sagsinto cavity 204, while a vacuum is applied to conform the softened glassto the shape that has been machined into molding surface 208. Towithstand the temperatures associated with this process, mold 200 may bemade of a heat-resistant material. As an example, the mold may be madeof high temperature steel or cast iron. To extend the life of the mold,the molding surface may be coated with a high-temperature material thatreduces interaction between the mold and the glass making up the glasscover, e.g., a chromium coating.

After removal from the mold and such post-molding processing as isdesired, e.g., annealing, the molded 3D glass cover is subjected to ionexchange strengthening. Various ion exchange techniques now known orsubsequently developed can be used depending on the specific performancerequirements of the 3D glass cover and the composition of the glassmaking up the cover. Examples of such processes can be found in U.S.Provisional Application No. 61/666,341 entitled “Methods For ChemicallyStrengthening Glass Articles” filed Jun. 29, 2012, to which U.S.application Ser. No. 13/923,837 claims the benefit of, the contents ofwhich are incorporated herein by reference. Examples of glasscompositions suitable for ion exchange strengthening can be found inU.S. Pat. Nos. 4,483,700, 5,674,790, 7,666,511, and 8,158,543; and U.S.Patent Application Publications Nos. 2009-0142568, 2011-0045961,2011-0201490, 2012-0135226, and 2013-0004758, the contents of which areincorporated herein by reference.

In broad outline, ion exchange strengthening involves treating theformed glass article by submersing it in a salt bath at an elevatedtemperature for a predetermined period of time. The process causes ionsfrom the salt bath, e.g., potassium ions, to diffuse into the glasswhile ions from the glass, e.g., sodium ions, diffuse out of the glass.Because of their different ionic radii, this exchange of ions betweenthe glass and the salt bath results in the formation of a compressivelayer at the surface of the glass which enhances the glass's mechanicalproperties, e.g., its surface hardness. The effects of the ion exchangeprocess are typically characterized in terms of two parameters: (1) thedepth of layer (DOL) produced by the process and (2) the final maximumsurface compressive stress (CS). Values for these parameters are mostconveniently determined using optical measurements, and commercialequipment is available for this purpose, e.g., instruments sold byFrontier Semiconductor and Orihara Industrial Company, Ltd.

As discussed above, in accordance with the present disclosure, it hassurprisingly been found that the ion exchange that occurs at theout-of-plane edge (or out-of-plane edges) of a 3D glass cover is a majordriver, indeed, in most cases, the major driver, of warp of the coverresulting from the IOX process. Although not wishing to be bound by anyparticular theory of operation, this effect can, in retrospect, beinterpreted in terms of bending moments. (Note that for ease ofreference, the following analysis uses the phrase “bending moment”instead of the more precise phrase “bending moment per unit length.”)

One of the simplest examples of a bending moment occurs in a long thinbar in which there is strain varying only in the thickness direction,taken to be the z-direction. The bending moment integral in this case isdefined by:

$\begin{matrix}{M = {E\; B{\int\limits_{{- h}//2}^{h/2}{{C(z)}{\mathbb{d}z}}}}} & (1)\end{matrix}$where E is the Young's modulus of the glass beam, B is the “latticedilation coefficient” (the factor that converts concentration ofexchanged ions into strain), C(z) is the concentration of the larger ionminus its value in the base glass, and the depth z varies from −h/2 onthe bottom surface to +h/2 on the top surface.

Using this definition, the final shape of the beam after ion exchangingthe top and bottom surfaces to produce the concentration profile C(z) isgiven by:

$\begin{matrix}{{w(x)} = {{{- 6}\frac{M}{E\; h^{3}}x^{2}} = {{- 6}\frac{B}{h^{3}}( {\int\limits_{{- h}/2}^{h/2}{{C(z)}z{\mathbb{d}z}}} ){x^{2}.}}}} & (2)\end{matrix}$

This result, which ignores any rigid body motion, is a valid descriptionof the z-component of displacement (called w here) along the centerlineof the beam (that is, through the middle) as a function of length alongthe beam where x=0 at the center of the beam. The final expression showsthat the bending of the beam is independent of Young's modulus E anddepends only on the lattice dilation coefficient B, the beam height h,and the concentration profile. A similar derivation for thermaldiffusion can be found in B. A. Boley and J. H. Weiner Theory of ThermalStresses, Dover Publications, 1988, p. 279 et seq. (hereinafter referredto as “Boley/Weiner”).

If the concentration profile is symmetric about the center z=0, then thebending moment integral of Eq. (1) is zero and there is no bending ofthe beam. If the concentration profile is asymmetric, e.g. as in thecase of float glass due to its asymmetry of manufacture, then thebending moment integral will be nonzero and the beam will take on aparabolic shape as given by Eq. (2). For example, if more ions areexchanged in the top half than in the bottom half, then the integral forM will be a positive number and the bending will be in a negative senseaccording to the minus sign in Eq. (2). This makes intuitive sensebecause where more ions enter there is greater expansion of the glasswhich causes the beam to bend away from this surface and toward theopposite surface.

In order to proceed to a 3D shape that bears some resemblance topractical parts of interest we consider an axisymmetric 3D case of adisk-shaped part that turns sharply at its perimeter. This is meant tosimplify realistic cases that typically have a rectangular shape (see,for example, FIGS. 1 and 16) and a bend radius ≧1 mm (see, for example,FIGS. 1, 16 and 17), whose magnitude may be a further variable of IOXwarping. As we show below, the warping of real 3D parts as a result ofion exchange of the glass edge is driven primarily by the same mechanismwe are about to describe; this mechanism involves expanding glass at theedge pushing to a larger size by bending the underlying part. Bending isdriven by a nonzero bending moment as in the simple example above.

The simplified geometry is shown in FIG. 3. Only a slice of the objectis shown, the full object being obtained by rotating this slice by 180°around an axis that extends vertically through its center. When theedges of this part are ion exchanged, the region of glass near thepoints of the arrows in FIG. 3 wants to expand relative to nearby glassthat is further from the surface. If the circumference of the ringcreated by the edge could be made larger it would relieve some of theelastic energy built up by the free strain from ion exchange. Forexample, imagine the part being pushed down onto the surface of a sphereso that it takes on a shape shown in FIG. 4.

If the part could take on the shape of FIG. 4, then the circumferencealong the edge of the glass actually becomes larger. Because this wouldrelieve some elastic energy, the part will tend to bend in this fashioneven though bending costs elastic energy. The resistance to bending ofthe entire part can be smaller than the energy gained by enlarging thecircumference at the edge depending on thickness and other details.

Quantitatively, the enlarging of the circumference is given as follows.If the original radius of the part prior to bending is a, then theoriginal circumference of the edge is:C ₀=2πa.  (3)

If the radius of curvature of the bent part is R (which is typicallymuch larger than a) then the new circumference of the in-plane glass(not the glass located at the arrow tips in FIG. 3, but the glasslocated at the right-angle turn) is, through second order:

$\begin{matrix}{C_{1} = {{2\pi\; R\;\sin\;\theta} = {{2\;\pi\; R\;\sin\frac{a}{R}} \approx {2\pi\mspace{11mu}{R( {\frac{a}{R} - {\frac{1}{6}( \frac{a}{R} )^{3}}} )}} \approx C_{0}}}} & (4)\end{matrix}$

This says the bending of the edge does very little to the originalcircumference of glass that was in the plane of the flat part. The glassat the very edge, however, gets the benefit of being out of the plane ofthe flat part by a little bit. Suppose it sticks up by an amount δ. Thenew circumference of the edge glass is then:

$\begin{matrix}{C_{1} = {{2\pi\;( {R + \delta} )\;\sin\;\theta}\; = {{2\pi\;( {R + \delta} )\;\sin\frac{a}{R}} = {{2\pi\;( {R + \delta} )( \frac{a}{R} )} = {C_{0} + {2\pi{\frac{a\;\delta}{R}.}}}}}}} & (5)\end{matrix}$

This says that bending of the part sticking out of the plane by δ addsadditional circumference that is proportional to δ. This provides asimple mechanism for relieving stress by taking advantage of slightlybending the part, but it is only relevant when a portion of the glass isout of plane and undergoes free strain from ion exchange. This also saysthat the edge region alone, the region just addressed in this analysis,is responsible for a driving force that tends to bend the disk-shapedpart in order to lower its elastic energy.

The driving force is related to elastic energy but can be expressed interms of forces to gain additional insight. FIG. 5 shows anotherdepiction of the original and warped shape. When the top edge is ionexchanged, if the edge were not constrained by being connected with therest of the saucer-shaped part, the edge would expand radially byroughly an amount:Δr=RBΔC  (6)where ΔC is the change in exchanged ion concentration and B is thelattice dilation coefficient mentioned in Eq. (1) above.

If unconstrained, the edge would expand freely in its own plane.However, because the edge is constrained by the vertical (or upward ingeneral) part of the glass adjoining it, and it is further constrainedby the rest of the part to which the vertical part is connected, therewill be a horizontal force and a bending moment generated. This providesa driving force to bend the part in order to accommodate the free strainof Eq. (6).

The above example can be examined in greater mathematical detail sincean axisymmetric plate with a concentration that varies only in theradial direction r and the thickness direction z with free boundaryconditions (and ignoring gravity) can be solved in certain cases (cf.Boley/Weiner, p. 389 et. seq.). The differential equation to be solvedis (cf. Boley/Weiner, Eq. (12.2.16)):

$\begin{matrix}{{{D{\nabla^{4}w}} = {\frac{1}{1 - v}{\nabla^{2}M}}}{where}} & (7) \\{D = {\frac{E\; h^{3}}{12( {1 - v^{2}} )}.}} & (8)\end{matrix}$

In these equations, w(r) is the vertical displacement of the part alongits midplane as a function of radius, D is sometimes called the bendingor flexural rigidity, E is Young's modulus (which will divide out so itdoes not enter the calculation), h is the original height of the part,and v is the Poisson ratio. M is the bending moment given by theintegral

$\begin{matrix}{{M(r)} = {B\; E{\int\limits_{{- h}/2}^{h/2}{{C( {r,z} )}z{\mathbb{d}z}}}}} & (9)\end{matrix}$where B is the lattice dilation coefficient converting concentration tostrain and C(r,z) is the concentration as a function of radius andvertical location. This is the same as Eq. (1) only with radialdependence.

Eq. (7) can be rewritten in the form

$\begin{matrix}{{\frac{1}{r}{\frac{\mathbb{d}\;}{\mathbb{d}r}\lbrack {r\frac{\mathbb{d}\;}{\mathbb{d}r}( {\frac{1}{r}\frac{\mathbb{d}\;}{\mathbb{d}r}( {r\frac{\mathbb{d}w}{\mathbb{d}r}} )} )} \rbrack}} = {\frac{12( {1 + v} )}{h^{3}}\frac{1}{r}{{\frac{\mathbb{d}\;}{\mathbb{d}r}\lbrack {r\frac{\mathbb{d}\;}{\mathbb{d}r}( {B{\int\limits_{{- h}/2}^{h/2}{{C( {r,z} )}z{\mathbb{d}z}}}} )} \rbrack}.}}} & (10)\end{matrix}$

The boundary condition along the outer radius of the part (r=a) for afree edge is given by (cf. Boley/Weiner, Eq. (12.4.26)):

$\begin{matrix}{{{D( {\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}} + {\frac{v}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} )} + \frac{M}{1 - v}} = {{{D\frac{\mathbb{d}\;}{\mathbb{d}r}( {\frac{1}{r}\frac{\mathbb{d}\;}{\mathbb{d}r}( {r\frac{\mathbb{d}w}{\mathbb{d}r}} )} )} + {\frac{1}{1 - v}\frac{\mathbb{d}M}{\mathbb{d}r}}} = 0}} & (11)\end{matrix}$

To make a simple and solvable example, we consider a uniformly expandingregion (representing an ion exchanged region, but simplified to uniformexpansion or constant concentration) expressed by the concentrationprofile:

$\begin{matrix}{{C( {r,z} )} = \{ {\begin{matrix}{C,} & {r_{1} < r < {a\mspace{14mu}{and}\mspace{14mu} z} \leq {{h/2} - \delta}} \\{0,} & {otherwise}\end{matrix}.}\; } & (12)\end{matrix}$

Notice that this introduces the same kind of asymmetry relative to themidplane as in the case studied earlier in that the ion exchanged regionis only near the top surface, not symmetrically near the bottom surfaceas well. This is what gives a nonzero bending moment. A cross-sectionalslice is sketched in FIG. 6. The region with uniform expansion is shownas a darker portion at the top outer edge. After some algebra the aboveequations are solved to give the following expression for warp vs.radius:

$\begin{matrix}{{w(r)} = {{- \frac{3}{2}}\frac{B\; C\;{\delta( {h - \delta} )}}{h^{3}}{\quad\lbrack {{( {1 - v} )( {1 - ( \frac{r_{1}}{a} )^{2}} )r^{2}} + {( {1 + v} ){\{ \begin{matrix}{0,{r < r_{1}}} \\{{( {r^{2} - r_{1}^{2}} ) - {2r_{1}^{2}{\ln( \frac{r}{r_{1}} )}}},{r \geq r_{1}}}\end{matrix} \rbrack.}}} }}} & (13)\end{matrix}$

Over most of the radius of the part, that is, inside r₁, the warp can besimplified to the form:

$\begin{matrix}{{w(r)} = {{- \frac{3}{2}}\frac{B\; C\;{\delta( {h - \delta} )}}{h^{3}}( {1 - v} )( {1 - ( \frac{r_{1}}{a} )^{2}} ){r^{2}.}}} & (14)\end{matrix}$

This shows that the originally flat part takes on a parabolic shape witha magnitude of warp that is proportional to δ (for small δ) andproportional to the free strain BC and approximately inverselyproportional to the square of the height h (one factor approximatelydivides out with the factor of (h−δ) in the numerator). Going exactly tothe edge, the warp is slightly altered from the parabolic shape, but theoverall scale is about the same. From the sign of the warp, we see thatpositive strain BC>0 creates a negative sense of warp, that is, the partbecomes concave down as shown elsewhere, e.g., FIG. 4.

The overall warp expressed as a single number can be taken to be themaximum vertical distance from the center to the new edge location. InEq. (13) this is just w(r=a), which gives the final result:

$\begin{matrix}{W = {{w( {r = a} )} = {{- \frac{3}{2}}\frac{B\; C\;{\delta( {h - \delta} )}}{h^{3}}{\quad\lbrack {{( {1 - v} )( {1 - ( \frac{r_{1}}{a} )^{2}} )a^{2}} + {( {1 + v} )( {( {a^{2} - r_{1}^{2}} ) - {2r_{1}^{2}{\ln( \frac{a}{r_{1}} )}}} )}} \rbrack}}}} & (15)\end{matrix}$

These analytic results provide insight into the roles of (1) bendingmoment (Eq. (9)) and (2) asymmetry of concentration in generating warp.To analyze more realistic cases, we switch from analytic methods tonumerical analysis. Specifically, we continue to treat basically thesame case but by using numerical finite element analysis, specifically,the ANSYS® commercial finite element software referred to above, we arefree to study more complex geometries and to use a realisticrepresentation of the concentration profile.

FIGS. 7-9 show the final state of warp as predicted/estimated by afinite element analysis for the same disk-shaped object studied above.The straight lines in these figures show where the glass was before ionexchange. FIG. 9 shows the concentration profile superimposed on thedistorted geometry (see the uppermost part of the distorted geometry).It is important to note that in this calculation only the very edge wasion exchanged, yet due to its impact on bending moments, this was enoughto drive all the distortion shown. This surprising result is cruciallyimportant for capturing the bending and warping of a part resulting fromion exchange.

With the importance of ion exchange on the edge of the part having beenestablished, we now turn to examples from finite element analysisshowing various trends in warp behavior with changes in part geometry.FIG. 10 shows the variables under study.

FIG. 11 shows results for angle α of 10° (left panels) and 90° (rightpanels) with 3 cases each: all surfaces ion exchanged (top), allsurfaces exchanged except the top edge (middle), and only the top edge(bottom). On the left, with smaller α, it is very clear that the warpingeffect is dominated by ion exchanging only the edges (bottom leftpanel). Practical cases of interest are usually closer to this low anglebending case; hence we focus on the edge exchange as a critical driverfor out-of-plane warp. In the case where the edge is exactly at a rightangle to the rest of the part, as shown in the right panels, additionalterms come into play so that the full vertical part and not just itsedge contribute to warping. Even in this case, ion exchange of the edgeis responsible for about a third of the warp.

As α varies from very small angles with a nearly flat part to 90° thereis an interplay of effects so the total amount of warp is not monotonicin α. Results are shown in FIG. 12. At very small α we can explain thetrend because at α=0 there is no longer any asymmetry and no drivingforce for warp. At low α, warp grows proportional to α. At somewherenear 10° another effect arises: as α grows larger the moment arm for thebending moment to bend the part becomes smaller (the outer radius of thepart is growing smaller as α increases) so the trend reaches a maximumand turns around. Additional subtleties could be examined but the mainpoint is that multiple considerations are at work to create a final partshape and a carefully constructed and accurate model is critical togetting useful results.

When β=90° we have the edge we have been discussing up to this point;other values of β introduce a bevel as shown in FIG. 10. The trend ofoverall warp with β is given in FIG. 13. Tilting away from about 90° ineither sense increases the warp. This is believed to be a result of theincreased surface area afforded by the bevel which allows more ions toexchange in the sample, driving a larger strain or a larger overall B·Cas mentioned in the analytic model.

The length of the out-of-plane feature has a non-monotonic impact onwarp similar to that of angle α. Results are shown in FIG. 14, where“curve length” represents the length of the perimeter portion of thepart. When the length of the out-of-plane curve approaches 0 then onceagain there is no asymmetry of geometry and no bending moment from ionexchange and no driving force for warp, so the warp must go to zero as xapproaches 0. At a length of around 2 mm in this example the warpreaches a maximum value and then starts to fall again. This is believedto be a result of additional rigidity provided by a longer out-of-planecurve. At some point with large enough x the out-of-plane curve canbecome rigid enough to prevent the flat base from warping.

The effects of flat length are shown in FIG. 15. From Eq. (15) we expectthe overall warp to rise roughly with the square of the radius a (or thesquare of the flat length in this case). As can be seen in FIG. 15, themain trend is parabolic as expected from the analytic model.

With the above understanding of the balance of forces, bending moments,and ion exchange-induced stresses, we now turn to the practical problemof predicting/estimating the warping of 3D glass covers as a result ofion exchange strengthening. Having an accurate model for this effect, wethen subtract off the effect from the original mold (e.g., an originalmold which is identical to the target shape), so that after ion exchangewarping the final shape agrees with the target shape. To manage 3Dshapes using finite element analysis, it is convenient to usewell-established computer programs. Commercial and open-source softwarepackages are usually designed to calculate thermal stresses and thermalwarp and not ion exchange stresses or ion exchange warp, so someadditional understanding is needed in order to create a thermal problemthat mimics the details of the ion exchange problem.

In accordance with the present disclosure, this is done using amathematical analogy. The mathematical analogy betweenconcentration/stress and temperature/stress exploits the fact that bothconcentration and temperature obey the same diffusion equation. In threedimensions, the governing equation for mass diffusion is:

$\begin{matrix}{\frac{\partial C}{\partial t} = {D\lbrack {\frac{\partial^{2}C}{\partial x^{2}} + \frac{\partial^{2}C}{\partial y^{2}} + \frac{\partial^{2}C}{\partial z^{2}}} \rbrack}} & (16)\end{matrix}$for the case of constant diffusivity D, where C represents concentrationof the diffusing species. The three-dimensional boundary conditions forthe cases of interest areC(x,y,z,t=0)=C _(base)  (17)at all points (x,y,z) inside the sample at the initial time andC(x,y,z,t)=C _(surf)  (18)on all the surfaces at all times.

When we say “ion exchange at the edge must be included in thecalculation” we mean that the C_(surf) boundary condition must beapplied at the thin edge that has been discussed previously and which isshown in FIG. 9 for the saucer example. Moreover, a sufficiently finemesh needs to be used to model the edge. FIGS. 16 and 17 show examplesof suitable meshes for use in modeling 3D glass covers. As can be seenin these figures and, in particular, in FIG. 17, the meshing for edge103 is finer than that used for the cover's central portion 101 and forthe major parts of its perimeter portion 102. A suitable mesh spacingfor the edge is in the range of 5-10 microns.

By Saint-Venant's Principle one would have argued that edge effectsshould not have much impact far from the edges, and one would haveignored ions entering this small region. Specifically, instead of usingthe above boundary condition, one would simply define the edge asimpermeable to ions. However, as shown above, the edges introduce anunbalanced bending moment that becomes primarily responsible for theoverall ion exchange-induced warping of the part, and thus cannot beignored.

In three dimensions, local “free strains” (before elastic relaxation orbefore incorporating material continuity or compatibility) are equal toB·C for lattice dilation coefficient B. (Recall B is the coefficientthat converts concentration to strain.) Three-dimensional stresscalculations use these initial strains along with stress boundaryconditions and compatibility conditions. The final 3D stresses andstrains are calculated based on these initial conditions and boundaryconditions using the techniques employed with thermally induced stressesand strains.

In three dimensions, the governing equation for thermal transfer (orequivalently, heat diffusion) is:

$\begin{matrix}{\frac{\partial T}{\partial t} = {{\frac{k}{\rho\; C_{p}}\lbrack {\frac{\partial^{2}T}{\partial x^{2}} + \frac{\partial^{2}T}{\partial y^{2}} + \frac{\partial^{2}T}{\partial z^{2}}} \rbrack} + \frac{Q}{\rho\; C_{p}}}} & (19)\end{matrix}$where T is the temperature, k is the thermal conductivity, ρ is thedensity, C_(y) is the specific heat capacity at constant pressure, and Qis a source of heat added per time per unit volume. This is the equationthat is solved by commercial finite element software such as the ANSYS®software referred to above.

Typical boundary conditions could assign an initial temperature that isuniform throughout the body and a surface temperature that is fixedthroughout the time evolution of the calculation. Free strains are givenby the coefficient of thermal expansion α times the temperature, α·T.Temperature plays the same role as concentration and coefficient ofthermal expansion α plays the role of lattice dilation coefficient B.The three-dimensional boundary conditions are handled by replacingconcentration with temperature.

To go further, Eq. (19) needs to be converted into Eq. (16). There isnothing in the ion exchange problem analogous to a heat source so in thethermal problem we set Q=0. Next we replace the quotient k/(ρC_(p)) by asingle constant known as the thermal diffusivity. We can set k, thethermal conductivity in the thermal problem, exactly to D, thediffusivity in the mass transfer problem, by setting ρC_(p)=1. Giventhat Q=0, the actual values of ρ and C_(p) are irrelevant but they mustbe chosen so as to keep ρC_(p)=1. The relationships between physicalconstants of the mass transfer and thermal problems are summarized inTable 1. The values of the physical properties listed in Table 1 can bereadily determined for any particular glass by skilled persons usingmeasurement techniques known in the art.

The analogy is completed by recognizing that (1) concentration andtemperature obey the same differential equation, (2) the same mechanicalconstants (Young's modulus, Poisson ratio) can be used in both kinds ofproblem, (3) the lattice dilation coefficient plays the same role forconcentration that is played by the thermal expansion coefficient withtemperature, and (4) the boundary conditions for a concentration problemcan also be taken to be boundary conditions for a thermal heat flowproblem. Thus, one may set temperature exactly equal to concentration,use Table 1 to see the correspondence of physical properties, and thencalculate concentration-derived stresses using an existing thermalmodel. This allows convenient use of finite element software that iswritten for the purpose of thermal stress modeling.

In this way, 3D glass covers having rectangular configurations like thatshown in FIG. 1 have been successfully modeled using the ANSYS® finiteelement thermal modeling software. Through this modeling, it has beenfound that when all the surfaces are ion-exchanged, higher warp isobserved along the diagonal axis as compared to the short and long axes.As a further example, a rectangular 3D glass cover having a perimeterportion which only extended upward from the long sides of the rectangle(i.e., a sled-like structure) was studied. In this case, the expansionof the edge causes high warp along the ends of long axis.

FIG. 18 illustrates the application of the above techniques to providemold contour correction values which achieve improved dimensionaltolerances of 3D glass covers. As shown in this figure, the methodincludes the following steps: (1) solve for IOX diffusion using thermalanalogy approach on edges and surfaces; (2) calculate predicted shapedeviation (warp) after IOX diffusion from given target (CAD) design; (3)inverse the warp values to get corrected values for the mold's moldingsurface; and (4) produce a molding surface having the corrected values.Once the mold is machined with the corrected values, the warp valuesafter IOX of 3D glass covers made with the mold are essentiallynegligible. Note that as discussed in Example 2 below, in some cases, itmay be desirable not to apply the full IOX correction to the moldsurface, in which case, after ion exchange, the covers will have someresidual IOX warp, but less than they would have had with an uncorrectedmold (see, for example, FIGS. 20 and 21 discussed below).

Typically, IOX of a 3D glass cover results in dome-shaped warp whereinwhen looking at the concave side of the shape, the center lifts up andedges move down. Accordingly, the inverse values used to correct themold will typically result in a mold surface that has an inverse domeshape, i.e., the portion of the mold that produces the flat or nearlyflat central portion of the 3D glass cover is not flat. However, theas-molded 3D glass cover with its inverse dome-shaped contour acquiredfrom the mold becomes flat or nearly flat after IOX, as is desired. Inaddition to warp, like with 2D parts, IOX of 3D glass covers alsoresults in an overall increase in part size.

The mathematical procedures described above can be readily implementedusing a variety of computer equipment, including personal computers,workstations, mainframes, etc. Output from the procedures can be inelectronic and/or hard copy form, and can be displayed in a variety offormats, including in tabular and graphical form. Software code,including data input routines for commercial software packages, can bestored and/or distributed in a variety of forms, e.g., on a hard drive,diskette, CD, flash drive, etc.

Molding of 3D glass articles in accordance with the present disclosurecan be performed using equipment now known or subsequently developed.Likewise, for ion exchange treatments, bath solutions that are now knownor subsequently developed can be used. Along the same lines, the glassarticles can have a variety of compositions suitable for ion diffusionprocessing now known or subsequently developed.

Without intending to limit it in any manner, the invention will befurther illustrated by the following examples. The glass used in theseexamples was Code 2317 glass, commercially available from Corning, Inc.(i.e., Corning Gorilla® glass). This representative, ion-exchangeableglass was also used in the simulations of FIGS. 7-15.

EXAMPLE 1

This example illustrates the importance of including the effects of ionexchange through the edge of a 3D glass cover when predicting IOX shapechanges.

FIG. 19 shows two warp predictions, one (the upper portion of thefigure) that included ion exchange through the edge of the 3D glasscover and the other (the lower portion of the figure) that did not. Inthis figure, the reference number 105 represents the target shape, whilethe reference numbers 106 and 107 represent the predicted shapes withand without edge IOX, respectively.

As can be seen in FIG. 19, failure to include the effects of edge IOXresults in a substantial underestimate of IOX warp, the magnitude of thepredicted/estimated warp without including edge IOX being ˜14 microns,while with the effects of edge IOX included, it increases to ˜130microns. Consequently, mold compensation based on predicted shape 107would result in manufacture of 3D glass covers with substantialdeviation from CAD after IOX that would be unacceptable to customers forsuch covers, while mold compensation based on predicted shape 106 wouldresult in glass covers that would meet customer specifications.

EXAMPLE 2

This example illustrates an application of the process steps of FIG. 18.In particular, the example compares a predicted shape for a 3D glasscover made with a corrected (compensated) mold versus the predictedshape for the same cover made with an uncorrected mold.

The results are shown in FIGS. 20 and 21, where FIG. 20 shows thebenefits of mold correction for the perimeter portion of the glass cover(identified by the variable “curve length” in this figure) and FIG. 21shows the benefits for the central portion (identified by the variable“flat length” in this figure). The “without correction” data assumes the3D glass cover was built with a mold whose molding surface matched thetarget shape, while the “with correction” data assumes the part wasbuilt with a mold whose molding surface was corrected to take account ofIOX warp, including IOX warp arising from ions passing through the edgeof the part.

It should be noted that the “with correction” data is for a mold thatwas not fully corrected for IOX warp. This data thus illustrates anembodiment of the present disclosure wherein predicted warp is notcompletely cancel out when modifying a mold, but less than the fullpredicted correction is made to accommodate other considerations thatmay need to be taken into account in a mold design, e.g., the cost ofmachining complex surfaces and/or thermal relaxation considerations.

Both the “without correction” and the “with correction” calculationsincluded the effects of edge IOX, the difference between the data beingthe “as-molded” shape at the beginning of the IOX process, the “withoutcorrection” as-molded shape being the target shape and the “withcorrection” as-molded shape being the target shape with the partialcorrection for IOX warp.

As can be seen in FIGS. 20 and 21, the post-IOX shapes of 3D glasscovers can be substantially improved using the mold correctiontechniques disclosed herein. For example, quantitatively, the data ofFIGS. 20 and 21 show a warp reduction after performing IOX from about 90microns to about 10 microns.

EXAMPLE 3

This example further illustrates the process steps of FIG. 18. Inparticular, the example compares a 3D glass cover made with a corrected(compensated) mold versus the same cover made with an uncorrected mold.

FIG. 22 shows the shape change resulting from IOX for a dish shape partwhose dimensions were 110 mm×55 mm×2 mm. The measured as-formed shapefor the uncorrected mold, i.e., the mold whose mold surface correspondedto the target shape, is shown by curve 108 in FIG. 22. As can be seen,the maximum magnitude of the flatness deviation of the central portionof the as-molded part was less than 20 microns (the vertical axes inFIGS. 22 and 23 are in millimeters). After IOX, the part warped stronglyas shown by curve 109, the maximum magnitude of the flatness deviationof the central portion now being greater than 80 microns.

Thereafter, the mold was corrected in accordance with the process ofFIG. 18, i.e., based on the predicted IOX warp for the part, and againused to make a 3D glass cover. The results of measurements on theas-molded (as-formed) part and the part after IOX are shown in FIG. 23,where curve 110 shows the shape of the as-molded part and curve 111shows the shape of the post-IOX part. As can be seen, by correcting themold, the flatness of the central portion of the as-molded cover wasmade worse, i.e., the maximum magnitude of the flatness deviation wasnow 90 microns, but the flatness of the central portion after IOX wasnow better, i.e., the maximum magnitude of the flatness deviation wasnow 40 microns.

Further experiments were performed in which compensated molds were usedto produce 3D glass covers of various configurations. Cases were studiedin which the correction removed some or essentially all warp at thecenter of the part after IOX. The second and third columns of Table 2set forth the predicted residual IOX warp at the center of the part(“model predictions”) and the measured warp at that location(“experimental measurements”). As can be seen, the predicted andmeasured values correspond closely thus demonstrating the ability of themold compensation procedures disclosed herein to control IOX warp of 3Dglass covers.

A variety of modifications that do not depart from the scope and spiritof the invention will be evident to persons of ordinary skill in the artfrom the foregoing disclosure. The following claims are intended tocover the specific embodiments set forth herein as well asmodifications, variations, and equivalents of those embodiments.

TABLE 1 Analogous Property Dimensions thermal value Young's Modulus EMPa Same Poisson ratio ν Same Nominal surface concentration (C_(surf))mole % Temperature Nominal initial concentration C_(base) mole % Initialtemperature Lattice dilation constant B 1/(mole %) CTE Diffusivity Dm²/sec Thermal diffusivity k/(ρC_(p)) Density ρ and specific heat C_(p)NA ρC_(p) = 1

TABLE 2 Center Warp Values in Microns Glass Model Experimental CoverShape Predictions Measurements Circular Dish 13 17 Rectangular Dish 1 9299 Rectangular Dish 2 53 42 Rectangular Dish 3 70 60 Rectangular Dish 4*75 63 *Bend radius on only two sides.

What is claimed is:
 1. A method of making a glass cover, said glasscover having a target three-dimensional shape which comprises a planarcentral portion and a perimeter portion which (i) borders at least partof the planar central portion and (ii) extends out of the plane of theplanar central portion to provide the glass cover with threedimensionality, said perimeter portion having a perimeter edge, saidmethod comprising: (I) providing a mold for forming the glass cover,said mold having a molding surface; (II) producing the glass cover usingthe mold of step (I); and (III) ion exchange strengthening the glasscover produced in step (II); wherein the method is characterized by: (a)the molding surface of the mold of step (I) is produced based at leastin part on computer-implemented modeling which predicts/estimateschanges to the target three-dimensional shape that will result when theion exchange strengthening of step (III) is performed, and (b) saidcomputer-implemented modeling includes modeling the effects of ionexchange through the perimeter edge.
 2. The method of claim 1, whereinin said computer-implemented modeling, the effects of ion exchangethrough the perimeter edge are included in the modeling through aboundary condition at the perimeter edge which permits ion permeationthrough the edge.
 3. The method of claim 2, wherein in saidcomputer-implemented modeling, the boundary condition at the perimeteredge specifies a constant ion concentration at the edge.
 4. The methodof claim 2, wherein in said computer-implemented modeling: (i) iondiffusion is treated as thermal diffusion; and (ii) based on thattreatment in the modeling, the boundary condition at the perimeter edgeis treated as permitting heat flow through the edge.
 5. The method ofclaim 4, wherein in said computer-implemented modeling, the boundarycondition at the perimeter edge specifies a constant temperature at theedge.
 6. The method of claim 1, wherein the molding surface is producedbased at least in part on the inverse of the changes to the targetthree-dimensional shape predicted/estimated by the computer-implementedmodeling.
 7. The method of claim 1, wherein in said computer-implementedmodeling a mesh is employed and the mesh has a size at the perimeteredge in the range of five to ten microns.
 8. The method of claim 1,wherein in said computer-implemented modeling, ion exchange through theperimeter edge results in predicted/estimated changes to the targetthree-dimensional shape of a larger magnitude than ion exchange throughthe remainder of the outer surface of the glass cover.
 9. The method ofclaim 1, wherein the glass cover produced in step (II) is for a portableelectronic device.